Base \(\Q_{23}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(12\)
Galois group $C_{13}:C_6$ (as 13T5)

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Defining polynomial

\(x^{13} + 23\) Copy content Toggle raw display


Base field: $\Q_{23}$
Degree $d$: $13$
Ramification exponent $e$: $13$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{23}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 23 }) }$: $1$
This field is not Galois over $\Q_{23}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{23}$
Relative Eisenstein polynomial: \( x^{13} + 23 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{12} + 13z^{11} + 9z^{10} + 10z^{9} + 2z^{8} + 22z^{7} + 14z^{6} + 14z^{5} + 22z^{4} + 2z^{3} + 10z^{2} + 9z + 13$
Associated inertia:$6$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_{13}:C_6$ (as 13T5)
Inertia group:$C_{13}$ (as 13T1)
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$13$
Wild slopes:None
Galois mean slope:$12/13$
Galois splitting model:Not computed